Problem 64 - Calculus of a Single Variable

Question

Sketch the graph of an arbitrary function \(f\) that satisfies the given condition but= does not satisfy the conditions of the Mean Value Theorem on the interval \([-5,5] .\) \(f\) is not continuous on \([-5,5]\)

Step-by-Step Solution

Answer

The graph of \(f\) is a straight line sloping downwards for \(x<0\) and upwards for \(x>0\), with a break at \(x=0\) to create discontinuity in the interval \([-5,5]\).

Step 1: Define discontinuity

A function is discontinuous when its graph has a break, hole or gap. Thus, sketching a discontinuous function requires including one of these features. Let's say \(f(x)\) has a break at \(x=0\).

Step 2: Sketch the graph

We can then sketch the graph of \(f\) in such way that it includes a break at \(x=0\). For \(x<0\), the graph could be a straight line sloping downwards from left to right and for \(x>0\), a straight line sloping upwards from left to right. These two lines should not meet at \(x=0\) to create a discontinuity.

Step 3: Violate mean value theorem

The Mean Value Theorem requires the function to be continuous over the interval \([-5,5]\). However, the sketch clearly shows a discontinuity at \(x=0\), and thus the function cannot satisfy the theorem.

Key Concepts

Discontinuity in Functions

Understanding discontinuity in functions is crucial for comprehending why certain theorems, such as the Mean Value Theorem, may not apply. Discontinuity occurs when a function exhibits sudden breaks, jumps, or holes in its graph. These are the points at which the function is not defined or has an abrupt change in value.

For instance, if you have a function that for all points less than zero follows one rule, and for all points greater than zero follows another, without ever meeting at zero, you've created a discontinuity at that point. Think of it as trying to draw the function without lifting your pencil, and yet you find you must – that's where the function is discontinuous. There are several types of discontinuities, such as

• Point discontinuity: Where the function has a hole at a certain point.
• Jump discontinuity: Where there is a sudden vertical leap in the function values.
• Infinite discontinuity: Where the function values head off towards infinity at a point.

Sketching Graphs of Functions

When it comes to sketching graphs of functions, visualizing how the function behaves over its domain is key. To sketch a graph, follow the basic behaviors of the function, noting increases, decreases, and constant regions. When a function is unable to satisfy certain theorems due to discontinuities, these interruptions should be clearly indicated.

In the step-by-step solution, a function with a break at zero was proposed. To sketch this, you begin with one part of the function before the break, say a descending line for negative values of x, and then, after a distinct gap to symbolize the break, continue with the second part after zero, like an ascending line for positive values of x. Always ensure to

Include Key Features

These can range from intercepts, asymptotes, maxima, and minima to intervals of increasing and decreasing behavior, and, importantly, points of discontinuity.

Continuous Functions

Continuous functions are the lifeblood of calculus. In simple terms, a function is continuous over an interval if you can draw it without lifting your pencil. No breaks, holes, or jumps – smooth sailing all the way. For a function to satisfy the conditions of the Mean Value Theorem, it must be continuous over the closed interval in question.

A continuous function ensures that for every point within the interval, there is no ambiguity about the function's value. It behaves nicely and as expected. This continuity is essential for the Mean Value Theorem, as it relies on the function being well-behaved to guarantee the existence of a tangent parallel to the average slope of the function over an interval. This captures the principle that within any continuous sweep, there is a moment where the instantaneous rate of change matches the average rate of change. If a function isn't continuous, there's no guarantee you can find such a point, as the sudden jumps or holes disrupt the needed smooth transitions.